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  "body": "\n## Definition\n\nLet $K_1$ and $K_2$ be disjoint oriented closed curves in $\\R^3$, each the image of a smooth or piecewise-linear embedding of the circle.\nThe group $H_1(\\R^3 \\setminus K_2; \\Z)$ of [[singular homology|M0Q5CFJ7ZH]] is infinite cyclic.\nIt has a preferred generator $\\mu$, the class of a small circle that bounds a disc meeting $K_2$ once.\nThis circle is oriented by the right-hand rule with respect to the orientation of $K_2$.\nThe *linking number* $\\operatorname{lk}(K_1, K_2)$ is the integer $m$ such that the class of $K_1$ in $H_1(\\R^3 \\setminus K_2; \\Z)$ equals $m \\mu$.\n\n## Remarks\n\nEquivalently, take a regular diagram of the [[link|1A5WRCNTM3]] $K_1 \\cup K_2$ and give each crossing between $K_1$ and $K_2$ the sign $+1$ or $-1$ by the right-hand rule.\nThen $\\operatorname{lk}(K_1, K_2)$ is half the sum of these signs.\nIn this form one sees that $\\operatorname{lk}(K_1, K_2) = \\operatorname{lk}(K_2, K_1)$.\n\nExample: in the Hopf link, the two circles cross twice in a standard diagram, and both crossings have the same sign.\nSo the linking number is $\\pm 1$, the sign depending on the orientations.\n\nNon-example of linking: in the unlink of two circles, the diagram has no crossings between the components, so the linking number is $0$.\nThe converse fails: the Whitehead link has linking number $0$, but its two components cannot be pulled apart.\n\nFor smooth curves $\\gamma_1, \\gamma_2$ parametrising $K_1, K_2$, the Gauss integral gives\n$$\\operatorname{lk}(K_1, K_2) = \\frac{1}{4\\pi} \\oint_{\\gamma_1} \\oint_{\\gamma_2} \\frac{(\\gamma_1 - \\gamma_2) \\cdot (d\\gamma_1 \\times d\\gamma_2)}{\\abs{\\gamma_1 - \\gamma_2}^3}.$$\n",
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